| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Definite integral with logarithmic form |
| Difficulty | Moderate -0.3 This is a straightforward multi-part integration question requiring standard techniques: expanding brackets and integrating trigonometric functions in part (a), recognizing logarithmic forms in part (b)(i), and evaluating a definite integral with logarithm laws in part (b)(ii). All techniques are routine for P2 level with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\int(2\cos^2\theta - \cos\theta - 3)\,d\theta\) | B1 | |
| Attempt use of identity to obtain integrand involving \(\cos 2\theta\) and \(\cos\theta\) | M1 | |
| Integrate to obtain form \(k_1\sin 2\theta + k_2\sin\theta + k_3\theta\) for non-zero constants | M1 | |
| Obtain \(\frac{1}{2}\sin 2\theta - \sin\theta - 2\theta + c\) | A1 | |
| Total: | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate to obtain form \(k_1\ln(2x+1) + k_2\ln(x)\) or \(k_1\ln(2x+1) + k_2\ln(2x)\) | M1 | |
| Obtain \(2\ln(2x+1) + \frac{1}{2}\ln x\) or \(2\ln(2x+1) + \frac{1}{2}\ln(2x)\) | A1 | |
| Total: | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use relevant logarithm power law for expression obtained from application of limits | M1 | |
| Use relevant logarithm addition/subtraction laws | M1 | |
| Obtain \(\ln 18\) | A1 | |
| Total: | 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\int(2\cos^2\theta - \cos\theta - 3)\,d\theta$ | B1 | |
| Attempt use of identity to obtain integrand involving $\cos 2\theta$ and $\cos\theta$ | M1 | |
| Integrate to obtain form $k_1\sin 2\theta + k_2\sin\theta + k_3\theta$ for non-zero constants | M1 | |
| Obtain $\frac{1}{2}\sin 2\theta - \sin\theta - 2\theta + c$ | A1 | |
| **Total:** | **4** | |
## Question 7(b)(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate to obtain form $k_1\ln(2x+1) + k_2\ln(x)$ or $k_1\ln(2x+1) + k_2\ln(2x)$ | M1 | |
| Obtain $2\ln(2x+1) + \frac{1}{2}\ln x$ or $2\ln(2x+1) + \frac{1}{2}\ln(2x)$ | A1 | |
| **Total:** | **2** | |
## Question 7(b)(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use relevant logarithm power law for expression obtained from application of limits | M1 | |
| Use relevant logarithm addition/subtraction laws | M1 | |
| Obtain $\ln 18$ | A1 | |
| **Total:** | **3** | |7
\begin{enumerate}[label=(\alph*)]
\item Find $\int ( 2 \cos \theta - 3 ) ( \cos \theta + 1 ) \mathrm { d } \theta$.
\item \begin{enumerate}[label=(\roman*)]
\item Find $\int \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x$.
\item Hence find $\int _ { 1 } ^ { 4 } \left( \frac { 4 } { 2 x + 1 } + \frac { 1 } { 2 x } \right) \mathrm { d } x$, giving your answer in the form $\ln k$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2017 Q7 [9]}}